# formal series

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## 11—20 of 22 matching pages

##### 11: 28.31 Equations of Whittaker–Hill and Ince
Formal $2\pi$-periodic solutions can be constructed as Fourier series; compare §28.4: …
##### 12: 2.3 Integrals of a Real Variable
Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: …
##### 13: 2.7 Differential Equations
All solutions are analytic at an ordinary point, and their Taylor-series expansions are found by equating coefficients. … The radii of convergence of the series (2.7.4), (2.7.6) are not less than the distance of the next nearest singularity of the differential equation from $z_{0}$. … these series converging in an annulus $|z|>a$, with at least one of $f_{0}$, $g_{0}$, $g_{1}$ nonzero. Formal solutions are … Hence unless the series (2.7.8) terminate (in which case the corresponding $\Lambda_{j}$ is zero) they diverge. …
##### 14: 27.5 Inversion Formulas
If a Dirichlet series $F(s)$ generates $f(n)$, and $G(s)$ generates $g(n)$, then the product $F(s)G(s)$ generates …
27.5.6 $G(x)=\sum_{n\leq x}F\left(\frac{x}{n}\right)\Longleftrightarrow F(x)=\sum_{n% \leq x}\mu\left(n\right)G\left(\frac{x}{n}\right),$
27.5.7 $G(x)=\sum_{m=1}^{\infty}\frac{F(mx)}{m^{s}}\Longleftrightarrow F(x)=\sum_{m=1}% ^{\infty}\mu\left(m\right)\frac{G(mx)}{m^{s}},$
##### 15: 16.2 Definition and Analytic Properties
Throughout this chapter it is assumed that none of the bottom parameters $b_{1}$, $b_{2}$, $\dots$, $b_{q}$ is a nonpositive integer, unless stated otherwise. Then formallyWhen $p\leq q$ the series (16.2.1) converges for all finite values of $z$ and defines an entire function. … Then the series (16.2.1) terminates and the generalized hypergeometric function is a polynomial in $z$. … In general the series (16.2.1) diverges for all nonzero values of $z$. … Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via …
##### 16: 18.38 Mathematical Applications
In consequence, expansions of functions that are infinitely differentiable on $[-1,1]$ in series of Chebyshev polynomials usually converge extremely rapidly. …
###### Differential Equations
Linear ordinary differential equations can be solved directly in series of Chebyshev polynomials (or other OP’s) by a method originated by Clenshaw (1957). … However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s. …
##### 17: 2.9 Difference Equations
Often $f(n)$ and $g(n)$ can be expanded in seriesFormal solutions are … For asymptotic expansions in inverse factorial series see Olde Daalhuis (2004a). … $c_{0}=1$, and higher coefficients are determined by formal substitution. … The coefficients $b_{s}$ and constant $c$ are again determined by formal substitution, beginning with $c=1$ when $\alpha_{2}-\alpha_{1}=0$, or with $b_{0}=1$ when $\alpha_{2}-\alpha_{1}=1,2,3,\dots$. …
Formally, …
##### 19: Software Index
 Open Source With Book Commercial … 25.21(ix) Dirichlet $L$-series ✓ a ✓ ✓ …
• Open Source Collections and Systems.

These are collections of software (e.g. libraries) or interactive systems of a somewhat broad scope. Contents may be adapted from research software or may be contributed by project participants who donate their services to the project. The software is made freely available to the public, typically in source code form. While formal support of the collection may not be provided by its developers, within active projects there is often a core group who donate time to consider bug reports and make updates to the collection.

• Software Associated with Books.

An increasing number of published books have included digital media containing software described in the book. Often, the collection of software covers a fairly broad area. Such software is typically developed by the book author. While it is not professionally packaged, it often provides a useful tool for readers to experiment with the concepts discussed in the book. The software itself is typically not formally supported by its authors.

• ##### 20: 2.6 Distributional Methods
Inserting (2.6.2) into (2.6.1) and integrating formally term-by-term, we obtain … The fact that expansion (2.6.6) misses all the terms in the second series in (2.6.7) raises the question: what went wrong with our process of reaching (2.6.6)? In the following subsections, we use some elementary facts of distribution theory (§1.16) to study the proper use of divergent integrals. …